I donot understand why Taylor series is used here to derive pressure? And partial derivative.

Please explain in calculus way. I really need to understand this ..

(a) (b) Fig. 26 3P=0 This same result will also occur in the x direction, and since the change in pressure is zero, it indicates that the pressure remains constant in the horizontal plane. In other words, the pressure will only be a function of z, p = p(z}, and so we can now express its change as a total derivative. From Fig. 2-6b, 21": = O; ptAA)(p + difdz)M7(AA AZ) = 0 rip = was (2-4) The negative sign indicates that the pressure will decrease as one moves upwards in the uid, positive .2 direction. The above two results apply to both incompressible and compressible uids, and in the next two sections we will treat each of these types of uids separately. *This is the result of a Taylor series expansion about a point, for which we have omitted ' 1 32p 1 61p . the higher-order terms, 2 y: l ' - - and 22 E ' ' , ,because theyr W)\" 53.1 2 622 drop out as slyrt] and dzrO. Also, the partial derivative is used here because the pressure is assumed to be changing in every coordinate direction. ie. the pressure is assumed to be different at each point, and so p = pLIJ'g']